Geochemistry of Precambrian mafic dikes,central Bighorn Mountains,Wyoming, U.S.A.

Precambrian quartz dolerites and metadolerites of the central Bighorn Mountains form dikes that intrude a Precambrian metamorphic and igneous terrane typical of the Laramide uplifts of the middle Rocky Mountains. They have a restricted range of major- and trace-element compositions and are typical of basalts in the middle stages of tholeiitic fractionation. Fractionation in the direction of iron enrichment occurred by removal of plagioclase. Average element concentrations of the two groups are nearly identical to one another, are comparable to those in Archean metabasalts from numerous shield areas, and are intermediate between those of modern oceanic tholeiites and continental tholeiites. These average concentrations suggest a depth of magma generation and thickness of crust intermediate between those for the oceanic and continental environments.     

The effects of boundary condition asymmetries on the interplanetary magnetic field-moon interaction

Boundary condition asymmetries inherent in the solar wind flow past the Moon are included in a cylindrical model of the interplanetary magnetic field - Moon interaction. Numerical examinations of the sunward side response of this model are compared in the frequency domain with those of symmetrically excited spherical and cylindrical models and two characteristic differences are observed: the response of the asymmetric model is depressed at low frequencies due to magnetic diffusion around a conducting core, and is flattened at high frequencies because of the finite application time of the incident interplanetary magnetic field. The diffusion of field lines around the core is also evident in the time response of the model in the anti-solar cavity. The above features of the lunar response resulting from boundary condition asymmetries are shown to be evident in observational measurements.     

A semi-analytic theory for the motion of a lunar satellite

A semi-analytical solution to the problem of the motion of a satellite of the moon is presented. Perturbative effects which are considered include those due to the attraction of the moon, earth, and sun, the non-sphericity of the moon's gravitational field, coupling of lower-order terms, solar radiation pressure, and physical libration. Short-period terms and intermediate-period terms, terms with the period of the moon's longitude, are produced by means of von Zeipel's method; it is proposed to obtain the secular perturbations, and those depending only on the argument of perilune, by numerical integration of the equations of motions. The short-period terms and intermediate-period terms are developed up to second order, where first order is 10^–2. The secular perturbations and perturbations dependent on the argument of perilune are obtained to third order.     

Hubble’s Law Implies Benford’s Law for Distances to Galaxies

A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances from the Earth to galaxies are a reasonably good fit to the probabilities predicted by Benford’s law, the well known logarithmic statistical distribution of significant digits. The purpose of the present article is to give a theoretical explanation, based on Hubble’s law and mathematical properties of Benford’s law, why galaxy distances might be expected to follow Benford’s law. The new galaxy-distance law derived here, which is robust with respect to change of scale and base, to additive and multiplicative computational or observational errors, and to variability of the Hubble constant in both time and space, predicts that conformity to Benford’s law will improve as more data on distances to galaxies becomes available. Conversely, with the logical derivation of this law presented here, the recent empirical observations may be viewed as independent evidence of the validity of Hubble’s law.     

On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.     

Modified capacity design rule for columns in tall steel MRFs with linear viscous dampers within the framework of Eurocode 8

Seismic design codes enforce a set of capacity design rules for steel moment-resisting frames (MRFs) to promote a ductile sway plastic mechanism that involves plastic hinges in beams and column bases. Previous research showed that these capacity design rules may not be effective for tall steel MRFs with viscous dampers under strong earthquakes due to high axial forces in columns. To address this issue, steel MRFs with linear viscous dampers of different stories are designed according to Eurocode 8 along with using a slightly modified conservative capacity design rule. According to this rule, the axial force for the capacity design of a column in the force path of viscous dampers is calculated as the envelope of the axial force from the peak drift state, and, the axial force from the peak velocity state times a scale factor. This envelope axial force value along with the bending moment and shear force from the peak drift state are used to carry out the capacity design of the column by using the formulae of Eurocode 8, i.e. in the same way with a column of a steel MRF without dampers. Incremental dynamic analyses for 44 earthquake ground motions show that the modified conservative capacity design rule results in steel MRFs with viscous dampers that have plastic mechanisms similar to those of steel MRFs without dampers. Moreover, the proposed capacity design rule becomes stricter for buildings with more than 10 stories to address that available analysis methods for structures with dampers underestimate the peak damper forces in the lower stories of yielding tall steel MRFs. More work is needed to extend the findings of this work to the case of steel MRFs with nonlinear viscous dampers.     

Large prandtl number finite-amplitude thermal convection with maxwell viscoelasticity

Abstract

Nonlinear interactions of deviatoric stress components and the velocity field occur in all dynamic flows where convected elasticity is accounted for. By incorporating a linear Maxwellian constitutive relation (Oldroyd ‘B’ type) into a finite-amplitude convection model we quantify the magnitude of some of the effects of these nonlinear interactions. For viscoelastic flows the relevant nondimensional parameter is the ratio of viscoelastic constitutive relaxation time constant, λ₁, to the basic flow process time. The Rayleigh number, Ra, and the nondimensional ratio of λ₁ to thermal conduction time, τ_c, are part of the parameter space investigated. However, shorter basic flow time scales than that for thermal equilibration are of interest since most viscoelastic fluids have relatively small values of λ₁ The ratio of λ₁ to buoyant time [bcirc], or λ₁/[bcirc], is, therefore, a pertinent parameter. Using both lithospheric and aesthenospheric values for λ₁, the ratio appropriate to mantle convection is roughly bounded by O(1)[bcirc]>λ₁/[bcirc]>O(10^?6). Employing these bounds and computing low Rayleigh number time-dependent convective flows in a two-dimensional box, it is demonstrated that viscoelasticity has a negligible influence on quasi-steady heat transport even for λ₁/[bcirc]～O(1) For any time-dependent behavior with time scales as short, or shorter than, the buoyant time, [bcirc], viscoelasticity might be important to the local exchange of mechanical energy. The recoverable strain energy in the descending portion of the lithosphere is comparable to the local viscous dissipation. The magnitude of this recoverable component of shear is proportional to λ₁/[bcirc].     

Source parameters of mining-induced seismic events: An evaluation of homogeneous and inhomogeneous faulting models for assessing damage potential

Source parameter estimates based on the homogeneous and inhomogeneous source models have been examined for an anomalous sequence of seven mine-induced events located between 640 and 825 m depth at Strathcona mine, Ontario, and having magnitudes ranging betweenm _N 0.8 and 2.7. The derived Brune static stress drops were found to be similar to those observed for natural earthquakes (30 bars), whereas dynamic stress drops were found to range up to 250–300 bars. Source radii derived from Madariaga's model better fit documented evidence of underground damage. These values of source radii were similar to those observed for the inhomogeneous model. The displacement at the source, based on the observed attenuation relationship, was about 60 mm for three magnitude 2.7 events. This is in agreement with slip values calculated using peak velocities and assuming the asperity as a Brune source within itself (72 mm). By using Madariaga's model for the asperity, the slip was over 3 times larger than observed. Peak velocity and acceleration scaling relations with magnitude were investigated by incorporating available South African data, appropriately reduced to Canadian geophysical conditions. The dynamic stress drop scaled as the square root of the seismic moment, similar to reported results in the literature for crustal earthquakes. This behavior suggests that the size of the asperities responsible for the peak ground motion, with respect to the overall source size, follow distributions that may be similar over a wide range of magnitudes. Measurements of source rupture complexity (ranging from 2 to 4) were found to agree with estimates of overall source to asperity radii, suggesting, together with the observed low rupture velocities (0.3 to 0.6 ), that the sources were somewhat complex. Validation of source model appropriateness was achieved by direct comparison of the predicted ground motion level to observed underground damage in Creighton mine, located within the same regional stress and geological regime as Strathcona mine. Close to the source (<100 m), corresponding to relatively higher damage levels, a good agreement was found between the predicted peak particle velocities for the inhomogeneous model and velocities derived based on established geomechanical relationships. The similarity between asperity radii and the regions of the highest observed damage provided additional support for the use of the inhomogeneous source model in the assessment of damage potential.